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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 108900.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
108900.f1 | 108900ct4 | \([0, 0, 0, -193306575, -1034470170250]\) | \(154639330142416/33275\) | \(171894386673900000000\) | \([2]\) | \(14929920\) | \(3.2674\) | |
108900.f2 | 108900ct3 | \([0, 0, 0, -12124200, -16044040375]\) | \(610462990336/8857805\) | \(2859892858287011250000\) | \([2]\) | \(7464960\) | \(2.9208\) | |
108900.f3 | 108900ct2 | \([0, 0, 0, -2731575, -981945250]\) | \(436334416/171875\) | \(887884228687500000000\) | \([2]\) | \(4976640\) | \(2.7181\) | |
108900.f4 | 108900ct1 | \([0, 0, 0, -1234200, 516927125]\) | \(643956736/15125\) | \(4883363257781250000\) | \([2]\) | \(2488320\) | \(2.3715\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 108900.f have rank \(0\).
Complex multiplication
The elliptic curves in class 108900.f do not have complex multiplication.Modular form 108900.2.a.f
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.